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Write a square form matrix online calculator. Quadratic form and quadric. Homology, power of homology

Quadratic form.
Sign-definiteness of forms. Sylvester's criterion

The perceptual "quadratic" is immediately drawn to the dummy, which is tied here with a square (by another step), and even sooner the knowledge of the "schoos" and the same form. I went straight ahead :)

I welcome you to my new level, and in the quality of a secret rosy, I can see the shape in a smart line. in a linear form winter name one-sided 1st degree polynomial:

- as specific numbers * (Admittance, if you want one of them, seemingly from zero), A - changes, which can accept a significant value.

* Within the framework of the given, those will look only numbers .

With the term "one-sided" we also stuck at the level of single line systems, І in this vipadku vin maє na uvazi, but the polynomial doesn’t have a constant.

for example: - line form of two wintry

Now the form is quadratic. quadratic winter name one-sided 2nd level polynomial, skin dodanok yakogo take revenge abo the square of the wintry, abo boyfriend tvir zminnykh. So, for example, the quadratic form of two wintry eyes is:

Uwaga! The price is standard notation, but there is no need to do it! Unimportant to the "terrible" view, everything is simple here - subordinate rows of indices of constants signal about those changes that are included in the same thing:
- Tvir і (square) is in the tomb;
- here tvir;
- i here tvir.

- Immediately before the juvenile a rude pardon, if you spend "minus" at the officer, not reasonably, you should be brought before the end of the day:

Inodi create a "school" version of the design in dusi, altogether otherwise, inodi. Before the speech, respect, for the rest of us here, we were concerned about not talking about it, and remembering "easy writing" is more important. Especially, if the winter is bigger.

The first quadratic form of the three winners is to take revenge on the same number of members:

... why should there be multiple "dviyki" multipliers in the "changes" before? It’s handy, and soon I’ll become smart, why.

However gagging formula can be written down, її can be manually decorated with a "prostrate":

- respectfully vivchaєmo skinny line - nothing terrible here!

The quadratic form is to avenge the dodanks with the squares of the wintry and dodanks with the young creators (Div. combinatorial formula for dnan) ... More than nothing - some "self-explanatory ix" and some add-on constants (the form is not a quadratic heterogeneous polynomial of the 2nd degree).

Matrix notation of quadratic form

If the meaning is clearly discernible, the form can be taken both positively and negatively, and the same can be taken for a linear form - if you want one of the functions of the positive, then there is with value).

This form is called sign... And if everything is clear from the line form, then with the square form, you can go kudi tsikavishe:

In a whole zoosuilo, so a form is given that can accept the meaning of any sign, such a rank, the quadratic form can also be sign.

And maybe not but:

- be sure that it is not equal to zero overnight.

- for be-like vector, Krim zero.

І zagalі, like for be-like non-zero vector ,, then the quadratic form is called positively singing; well, then negatively singing.

And everything was good, the lesser value of the quadratic shape is visible only in simple butts, and the visibility is lost even with a small acceleration:
– ?

Can you let it go, but the form is positively marked, or is it right? Raptom іsnuyut value, for which won less than zero?

Na tsei rakhunok isnu theorem: that's all power numbers matrix quadratic form positive * , Then it is marked positively. If everything is negative, then everything is negative.

* In theory, it is brought that all the power numbers of the symmetric matrix dіysnі

Writable by the matrix of the induced form:
і s rіvnyannya we know її strong meaning:

Virishuєmo old good squarely:

, Means the form It is positively signified, so that if there are non-zero values, it is more than zero.

The method is very practical, ale є one great ALE. Already for the matrix "three by three", the shukati are powerful numbers - є busy dovge and inappropriate; from the high ymovirnistyu see the polynomial of the 3rd degree with the irrational roots.

Yak bootie? Isnu big simple way!

Sylvester's criterion

Hi, not Sylvester Stallone :) kutovі minori matrices. tse Visitors how to "grow" from the left upper kut:

and stay from them in the exactness to the matrix holder.

Now, vasne, criterion:

1) The quadratic form is indicated positively todi and only todi, if ALL kutovs of minors are greater than zero :.

2) The quadratic form is indicated negatively todi and only todi, if kutovi minori Signs, if the 1st minor is less than zero: “if it's a guy, if it's not paired.

Yaksho hocha b one kutoviy minor of the sign, then the form signs of change... As kutovі minori "that" sign, even in the middle of them є nilovі, then there are special vidados, which I will pick out the trochi more, for that, as we click more wider butts.

About the analysis of kutovі minori matrixes :

First of all, we should tell us about those whose form is not negatively signified.

visnovok: All kutovі minori more than zero, out, shape positively marked.

Є Difference from the method of power numbers? ;)

Writable by the matrix of the form z butt 1:

first її kutoviy minor, and the other , The stars of the vaping, which form are signified, so that in the fallowness of the meaning, you can accept both positive and negative meanings. Well, it’s so obvious.

Take the form і її matrix z butt 2:

here vzagalі do not rozіbratіnya without feeling. Ale z criterієm Sylvester we are all free:
Well, the form is definitely not negative.

, І definitely not positive (To that, all kutovі minori are guilty but positive).

visnovok: The form is marked.

Pink butt for independent solution:

butt 4

Preceding quadratic forms for definiteness of signs

Everything is smooth with cich butts (div. Kinets lesson) Sylvester's criterion may not be enough.

On the right, in the fact that you can see the "edge" of the vip non-zero vector, then the shape is non-negative, which is unpositive... In qih forms іsnu non-null vector, with yak.

Here you can set up such a "button accordion":

see outer square, At once Bachimo non-negativity form:, moreover, won’t cost zero for any vector with equal coordinates, for example: .

"Mirror" butt unpositive singing form:

and even more trivial butt:
- here the form is leading to zero for any vector, de - a sufficient number.

Yak viyaviti non-negativity or non-positive form?

For whom we will need an understanding head minor matrices. Golovny Minor - a price for a minor, of storage with elements, which stand on the overflow rows and 100% with the same numbers. So, the matrix has two head minors of the 1st order:
(The element is located on the cross of the 1st row and 1st stovpchik);
(The element is located on the cross of the 2nd row and the 2nd hundred),

і one head minor of the 2nd order:
- stacked with elements of the 1st, 2nd rows and 1st, 2nd hundredths.

The matrix "three by three" to the head minor, and here it is already possible to wave the biceps:
- three minors of the 1st order,
three minors of the 2nd order:
- folded from the elements of the 1st, 2nd rows and 1st, 2nd hundredths;
- stacked with elements of the 1st, 3rd rows and 1st, 3rd hundredths;
- folded from the elements of the 2nd, 3rd rows and the 2nd, 3rd century,
і one minor of the 3rd order:
- stacked with elements of the 1st, 2nd, 3rd rows and 1st, 2nd and 3rd rows.
zavdannya on the smart: write down all the heads of the matrixes .
Zviryaumosya in kintzi lesson and prodovzhuumo.

Schwarzenegger's criterion:

1) Nonulova * quadratic form is assigned non-negative todi and only todi, if ALL of the heads of the minors unavailable(More or less).

* In zero (virogenic) quadratic form, all performance is returned to zero.

2) The non-Nullian quadratic form with the matrix is assigned unpositive Todi and only Todi, if:
- head minors of the 1st order non-positive(Less or less back to zero);
- head minori of the 2nd order unavailable;
- head minori of the 3rd order non-positive(Pishlo cherguvannya);
…
- head minor of the th order not positive, Yaksho - unpaired abo non-negative, Yaksho is a guy.

If there is one minor opposite to the sign, then the form is signified.

We'll be surprised, yak pratsyuє criterion in the guided butts:

Warehouse form matrix, i pershu cherga numbered kutovі minori - is the raptom sign positively or negatively?

The recognition of the meaning does not meet the criteria of Sylvester, however, another minor NOT negative, І tse wiklikє need to revise the 2nd criterion (In the case of the 2nd criterion, there will be no visoning automatically, so that visnovoks will immediately be afraid of sign-shaped forms).

Head minors of the 1st order:
- positive,
head minor of the 2nd order:
- Chi is not negative.

In such a rank, ALL the heads of the minori are neutral, meaning, the form non-negative.

Writable by the form matrix For those, obviously, Sylvester's criterion is not viconano. Alle і of the other signs, they may not have been rejected (for that, the offense of the kutovs to the minors should be taken away to zero). To this, the criterion of positivity / non-positive is reconsidered. Heads of the 1st order:
- not positive,
head minor of the 2nd order:
- Chi is not negative.

By such a rank, according to Schwarzenegger's criterion (point 2), the form is not positive.

Now the all-over-the-counter tasks have been taken care of:

butt 5

Doslіditi a quadratic form on sign definiteness

I will give the form to embellish the order "alpha", which can be used for any design number. Ale tse zh tilki will be more cheerful, virishuєmo.

A spatka can be written by the form matrix, melodiously, richly, but also cemented head diagonal the efficiency is set for squares, and on the symmetrical mice - the half of the efficiency of the different "changes" of the creatures:

Numbered kutovі minori:

the third card holder I rozkriyu on the 3rd row:

When dealing with different applied problems, it is often brought up to the level of quadratic forms.

Viznachennya. The quadratic form L (, x 2, ..., x n) of the n minions is called the sum, the dermal term of which є or the square of one of the minions, or even two of the minions, who took the deeds of the performance:

L (, x 2, ..., x n) =

Allowance, but the efficiency of the quadratic form is the design of the number, moreover

The matrix А = () (i, j = 1, 2, ..., n), is composed of a number of factors, called a matrix of quadratic form.

The matrix notation has a quadratic form of the ma viglyad: L = X "AX, de X = (x 1, x 2, ..., x n)" - matrix-hundredpets of winter.

butt 8.1

Write down the quadratic form L (, x 2, x 3) = in the matrix view.

We know the matrix of the quadratic form. Some diagonal elements are equal to the coefficients for the squares of the wintry, to be 4, 1, -3, and the other elements are half of the common coefficients of the quadratic form. Tom

L = (, x 2, x 3) .

In the case of a non-virgin linear transformation X = CY, the matrix of the quadratic form of the swelling eye is: A * = C "AC. (*)

butt 8.2

You are given a quadratic form L (x x, x 2) = 2x 1 2 + 4x 1 x 2 -3. Know the quadratic form L (y 1, y 2), rejected from the given linear transformations = 2y 1 - 3y 2, x 2 = y 1 + y 2.

The matrix of the given quadratic form is A =, and the matrix of the linear transformation

C =. Otzhe, by (*) matrix of Shukan square form

And the form of ma viglyad is quadratic

L (y 1, y 2) = .

Slid means that when doing something in the distance, the form of a quadratic form can be easily simplified.

Viznachennya. The quadratic form L (, x 2, ..., x n) = be called canonical (or a canonical viewer), as all the features = 0 for i¹j:

L = , A її matrix є diagonal.

The theorem is valid.

Theorem. Even if the form is quadratic, it will be reduced to the canonical viglyad by the addition of a non-virgin linear transformation of the winter.

butt 8.3

Bring the quadratic form to the canonical form

L (, x 2, x 3) =

There is a list of visible squares when changing, efficiency when squares are displayed as zero:

Now we see a new square at the change, the efficiency at any given view of zero:

Otzhe, non-virgin line-up

to produce a given quadratic form to the canonical viglyad:

The canonical view of the quadratic form is not uniquely valued, since one and the same quadratic form can be reduced to the canonical view in many ways. However, otrimanі in different ways canonical form Mayut a number outlandish authorities... One of the ruling powers is formulated in viglyadi theorems.

Theorem (the law of inertia of quadratic forms). The number of additions with positive (negative) coefficients of a quadratic form does not lie in the way of reducing the form to its form.

Slide to mean that the rank of the matrix of the quadratic form according to the number of different types of zero of the coefficients of the canonical form and does not change in case of liner transformations.

Viznachennya. The quadratic form L (, x 2, ..., x n) is called positively (negatively) singing, as with all the meanings of the changes, for which one wants to be seen from zero,

L (, x 2, ..., x n)> 0 (L (, x 2, ..., x n)< 0).

So, by the way, Quadratic form є positively singing, and the form is negatively meaningful.

Theorem. For this purpose, the quadratic form L = X "AX was positively (negatively) singing, it is necessary and sufficient, but all powerful values, matrices A boules are positive (negative).

service recognition... Online calculator vicarist Hesse matrices and for the type of function (opucle is aborted) (div. butt). The decision is made in Word format. For the function of one change f (x), there are intervals of opacity and oppression.

Rules for introducing functions:

The function f (x) can be infinitely differentiated between two and only todly, if Hesse matrix the function f (x) with respect to x is positively (negatively) semidefinite for all x (div. points of the local extremums of the functions of the miners).

Critical points of the function:

if the Hessian is positive with respect to the values, then x 0 is the point of the local minimum to the function f (x),
if the Hessian is negative in the values, then x 0 is the point of the local maximum of the function f (x),
if the Hessian is not definite of sign (taking both positive and negative values) and non-virulence (det G (f) ≠ 0), then x 0 is the sided point of the function f (x).

Matrix value criteria (Sylvester's theorem)

positive value:

all diagonal elements of the matrix are positive;
All the guilty principals are positive.

For positively semidefinite matrices Sylvester's criterion to sound like this: The form is positively semi-determined todi and only todi, since all the head minors are unavailable. If the Hessian matrix at the points is positively semi-determined (all the head minors are not), then the point is minimum (prote, as the Hessian is semi-defined, and one of the minors is semi-defined, and then there is a possibility of additional and middle point.

Semi-definiteness is positive:

all diagonal elements are not;
All the heads are not used.

The head card holder is the head card holder for the head minor.

A square symmetric matrix of order n, elements of which є private old central functions of a different order, be called the Hesse matrix i mean:

In order for a symmetric boolean matrix to be positively assigned, it is necessary and sufficient, all the diagonal minor bullets are positive, tobto

for matrices A = (a ij) positive.

negative value.
For that, a symmetrical boolean matrix is negatively signified;
(-1) k D k> 0, k= 1, .., n.
In other words, in order for the boolean to be square negatively singing It is necessary and sufficient, but the signs of the cube minors of the matrix of the quadratic form were drawn, repaired from the minus sign. For example, for two winters, D 1< 0, D 2 > 0.

If the Hessian is semi-definite, then it can be a speck of a hump. Required additional updates, which can be carried out one at a time from the next options:

lower order... To be afraid to replace the winners. For example, for the function of two different values y = x, as a result, we can accept the function of one variable x. You can see the behavior of the function on the straight lines y = x and y = -x. In the first place, the function is in the last point of the minimum, and in the first point, the maximum (or navpaki), then the point is the saddle point.
Significance of the power of the Hessians. As well as all the meanings of the positive, the function in the preliminarily points of the minimum, as all the negative ones are the maximum.
Additional function f (x) in the vicinity of point ε. Change x to change to x 0 + ε. Further, it is necessary to bring the function f (x 0 + ε) as a single change ε, either more than zero (to the point x 0 to the minimum), or less than zero (to the point x 0 to the maximum).

Note... Know zvorotny hessian finish to know the ringing matrix.

Butt number 1. For those of the advanced functions є for the opuque or for the reduced ones: f (x) = 8x 1 2 + 4x 1 x 2 + 5x 2 2.
Decision... 1. We know the privacy of old.

2. Virishimo the system of priests.
-4x 1 + 4x 2 +2 = 0
4x 1 -6x 2 +6 = 0
otrimaєmo:
a) The first time in the family x 1 and presented in another family:
x 2 = x 2 + 1/2
-2x 2 +8 = 0
Stars x 2 = 4
The given value x 2 is given for the viraz for x 1. It is acceptable: x 1 = 9/2
Number of critical points on the road 1.
M 1 (9/2; 4)
3. We know the privacy of a different order.

4. The numerical value of the number of private ones of another order at the critical points M (x 0; y 0).
Numerical value for point M 1 (9/2; 4)

I will be the Hesse matrix:

D 1 = a 11< 0, D 2 = 8 > 0
Oscillations of diagonal minors may be different signs, then nothing can be said about the opacity or the oppression of the function.

Square shape L from n wintry people are called suma, a skin member of which is either a square of one of the worms, or a little bit of two wintry wines.

Vvazayuchi, scho in a quadratic form L In the same way, the given details of the members are introduced, and the following values are introduced for the performance of the form: the performance is meaningfully through, and the performance for the creation is through. So, it’s the function of the whole creation of meaning and through, so that the meaning we have introduced, allows for the fairness of equality. You can write a member now at the viglyadі

and the whole quadratic form L- at the viglyadi sumi all the members, de iі j in the same place one of the same take the value
from 1 to n:

(6.13)

Three features can be combined with a square matrix of order n; get your name matrix of quadratic form L, A її rank - rank tsієї quadratic form. Yaksho, zokrema, tobto matrix is non-virgin, then i is a quadratic form L be called non-virgin... So yak, then the elements of the matrix A, symmetrical in the head diagonal, equal to themselves, so that the matrix A - symmetrical... Back, for a symmetric matrix A n-th order, we can use the entire quadratic form (6.13) as n the winners, we have the elements of the matrix A with their own performance.

The quadratic form (6.13) can be represented in the matrix view, which was introduced in Section 3.2 of the multiple matrices. Apparently through X hundredpets, folds from winter

X is a matrix, which has n rows and one hundred. By transposing a qyu matrix, you can render a matrix , Folded in one row. The quadratic form (6.13) with the matrix can be written now in the view of the offensive:

Fair:

and the equivalence of formulas (6.13) and (6.14) has been established.

Write її in the matrix viewer.

○ We know the matrix of the quadratic form. Some diagonal elements are equal to the coefficients for the squares of the wintry, to be 4, 1, -3, and the other elements are half of the common coefficients of the quadratic form. Tom

. ●

Z'yasuєmo, like the change is a quadratic form with a non-virgin linear re-incarnation of wines.

Amazingly, if the matrices A and In such a way, if the temperature is assigned, then there is little parity:

(6.15)

Surely, if tvir AB is assigned, then it will be assigned tvir: the number of hundreds of matrices for the number of rows in the matrix. Matrix element i-th row i j-m stovpts, in the matrix of AV seams in j-th row i i-m hundredpts_. Winners to that sum of creations of all kinds of elements j- the rows of the matrix A and i-th hundredth matrix B, tobto dorіvnyu sumi creation of similar elements in a row j-th hundredth matrix i i first row of matrices. Tsim parity (6.15) brought.

Leave the matrix-hundred of the winners і knitted with lineage X = Cy, de C = ( c ij) є deyaka non-virogena matrix n th order. Todi is a quadratic form

abo , De.

The matrix will be symmetric, so as in respect of equality (6.15), which is obviously fair for any number of multipliers, and equal to the symmetry of matrix A, it is possible:

Otzhe, with a non-virgin linear transformation X = Cy, the matrix of the quadratic form swells to the eye

Respect. The rank of the quadratic form does not change when a non-virulent linear re-creation is made.

Butt. Given a quadratic form

Know the quadratic form, rejected from the given linear re-incarnations

, .

○ The matrix of the given quadratic form , And the matrix of the linear transformation ... Also, by (6.16) the matrix of the Shukan quadratic form

and the form of ma viglyad is quadratic. ●

When people are far away from the line, the form of the quadratic form can be easily simplified.

quadratic form be called canonical(Abo maє canonical view), Yaksho all її features i≠ j:

and її matrix є diagonal.

The theorem is valid.

Theorem 6.1... Even if the form is quadratic, it will be reduced to the canonical viglyad by the addition of a non-virgin linear transformation of the winter.

Butt. Bring the quadratic form to the canonical form

○ The list of visible squares in case of change, efficiency in case of square of what kind of display is zero:

Now you can see a square when changing, the efficiency when square is what you see from zero:

Otzhe, non-virgin line-up

to produce a given quadratic form to the canonical viglyad

.●

The canonical view of the quadratic form is not uniquely valued, since one and the same quadratic form can be reduced to the canonical view in many ways. However, otrimanі in different ways canonіchnі form a number of overwhelming powers. One of the ruling powers is formulated in viglyadi theorems.

Theorem 6.2.(The law of inertia of quadratic forms).

The number of additions with positive (negative) coefficients of a quadratic form does not lie in the way of reducing the form to its form.

For example, a quadratic form

yaku looking at the side. 131 butts brought to mind

it is possible to have it, having stuck in a non-virulent line of transformation

bring to mind

Yak bachimo, the number of positive and negative features (apparently, two or one) was saved.

Remarkably, the rank of the quadratic form is based on the number of values of zero of the canonical form factor.

quadratic form to be called positively (negatively) singing, as with all the meanings of the wintry, for which one wants to be seen from zero,

().

Understand the quadratic form. Matrix of quadratic form. Canonical view of quadratic form. Lagrange's method. Normal view of the quadratic form. Rank, index and signature of the quadratic form. A positive quadratic form. Quadrik.

Understanding the quadratic form: a function on the vector space, which can be set as a one-sided polynomial of another degree from the coordinates of the vector.

Square form n unavailable to be called a suma, a dermal dodanok such as є or a square of one of the cich are not homeless, or a little bit of two children who are not homeless.

Quadratic form matrix: The matrix is called the matrix of the quadratic form in the given basis. At times, if the field characteristic is not expensive 2, it is possible to consider that the matrix of the quadratic form is symmetric, that is.

Write a matrix of quadratic form:

already,

The vector-matrix form has a quadratic form of the ma view:

A, de

Canonical view of the quadratic form: A quadratic form is called canonical, since everything i.e.

If a quadratic form can be brought to a canonical form, with additional linear re-creation. On a practical level, you can use these methods.

Lagrange method : the last time the new squares were seen. Forward, yaksho

Let’s try to use the procedure in a quadratic form and so on. In the quadratic form, everything is al then, after the forefront re-implementation, there is only about the marriage of the opened procedure. So, yaksho, for example, then vvazhaєmo

Normal view of the quadratic form: Such a canonical quadratic form is called a normal quadratic form;

Rank, index and signature of the quadratic form: By the rank of the quadratic form A called the rank of the matrix A... The rank of the quadratic form does not change in the case of non-virulent re-incarnations of non-native ones.

The number of negative performances is called a negative index of the form.

The number of positive members in the canonical view is called the positive index of the inertia of the quadratic form, the number of negative members - the negative index. The recognition of both positive and negative indices is called the signature of the quadratic form.

The positive quadratic form is: Material quadratic form to be called positively singing (negatively meaningful), as if they are not equal, the speech meanings of the winners are immediately zero

. (36)

In general, the matrix is also called positively singing (negatively singing).

The class of positively singing (negatively singing) forms is a part of the class of non-positive (apparently non-positive) forms.

quadric: quadrik - n-mirnim hypersurface in n+ 1-vimіrnuyu space, set yak bezlіch zeros of baggage of another step. I should enter the coordinates ( x 1 , x 2 , x n+1) (in the Euclidean abo affine space), zalne rivnyannya kvadrik maє viglyad

The price can be rewritten more compactly in matrix values:

de x = ( x 1 , x 2 , x n+1) is a row vector, x T - transposition vector, Q- matrix size ( n+1) × ( n+1) (transfer, if there is only one non-zero element), P is a row vector, and R is a constant. Nibbles often look at the squares over the real complex numbers. The design can be expanded to a quadric in the design space, div. Lower.

Bigger, meaningless zeros of the system of polynomial rivnyans in the form of algebraic development. In such a rank, the quadric є (affine or projective) algebraic functions of another degree and codimension 1.

The re-development of the area and space.

The designation of the re-development of the area. Viznachennya ruhu. power to the ruk. Two kinds of rukhiv: rukh of the I kind and rukh of the II kind. Attach rukhiv. Analytical viraz to ruhu. Classification of rucks in the area (in fallowness due to the manifestation of unruffy points and invariable straight lines). A group of rucks in the area.

Viznachennya redevelopment of the area: Viznachennya. The re-creation of the area is called collapse(For displacement) areas. The re-development of the area to be called affinim If there are three points, if you lie on the same straight line, translate into three points, as well as lie on the same straight line and when there are several points, it is easy to find three points.

Viznachennya ruhu: the price of the re-creation of figures, when you pick them up, there are points between them. If two figuri are exactly one by one behind an additional ruch, then the figuri are the same, rivni.

Power to the Ruhu: Any kind of arrangement of the surface of the area either by a parallel transfer, or by a turn, any minimum arrangement of the area either by an axial symmetry, or an axial symmetry. Points that lie on a straight line, when rus, go over to points that lie on a straight line, and the order of their mutual expansion is maintained. When Rusі, there are kuti among the exchanges.

Two kinds of rukhiv: rukh of the I kind and rukh of the II kind: Rukhs of the first genus are those rukhs that take advantage of the organization of the bases of the figurines. The stench can be realized without interruptions.

Rukhs of a different genus - those rukhs that change the basis for the opposite. The stench cannot be realized without interruptions.

With the butts of the rucks of the first genus - transfer і turn about straight, and with the rucks of another genus - the central and mirror symmetry.

Composition of any number of rukhs of the first clan є rukh of the first clan.

The composition of the paired number of rukhivs of another genus is rukh of the 1st genus, and the composition of the unpaired number of rukhivs of the 2nd genus is rukh of the 2nd genus.

Attach rukhiv:parallel transfer. Nekhai a - danish vector. Parallel transfer to the vector a is called the imaged area on itself, when the skin point M is mapped to the point M 1, but the vector MM 1 is to the vector a.

Parallel transfer є collapsed, fragments is a representation of the area on itself, which is taken from the perspective. Actually, the whole area can be represented directly in the given vector and on the whole area.

Turn. Significantly on the area point О ( center turn) I is given a cut α ( kut turn). By turning the area near the point O on the cut α, we call the image of the area on itself, when the skin point M is displayed at the point M 1, where ОМ = ОМ 1 and the cut MOМ 1 road α. At the same time, the point is about to be lost on your own mind, i.e., to appear in itself, and all of the points turn around the point O in the same direction - for the year-old line or against the year-old line (on the small picture of the image).

Turn є with a fall, shards are a representation of the area on itself, when you see it.

Analytical viraz to ruhu: is analogous to the link, between the coordinates to the preimage and the image of the point of the eyeglass (1).

Classification of rucks in the area (in fallowness due to the manifestation of unkempt points and innovative straight lines):

The point of the square is invariable (unruly), which, when re-created, passes in itself.

Butt: With central symmetry Invariant - point to the center of symmetry. When turning invariant є point to the center of the turn. at axial symmetry Invariant - straight - all symmetry - the chain of straight invariable points.

Theorem: If the problem is not an indispensable invariant point, then it will be more invariably invariable.

Butt: Parallel transfer. It is really, straight, parallel to the directly invariable figure in general, if you do not need to buy from the invariable points.

Theorem: If you collapse like a promin, if you translate in yourself, then the whole collapse or the same transformation, or the symmetry is obviously straightforward to avenge the promin.

To that, for the obviousness of the invariable points, or figures, it is possible to carry out the classification of rukhiv.

Name the ruhu	Invariant points	Invariant direct
Rukh of the 1st kind.
1. - turn	(Center) - 0	dumb
2. The same transformation	all points of the area	all straight
3. Central symmetry	point 0 - center	all straight lines that pass through point 0
4. Parallel transfer	dumb	all straight
Rukh of the II kind.
5. Axial symmetry.	without points	all symmetry (straight) all straight

Group of rukh_v area: In geometry, an important role is played by groups of self-aligning figures. If you are a figure on the area (or in the open space), then you can glance at all quietly in the area (or open space), with which figure you can move in yourself.

Tse bezlich є group. For example, for a single-sided tricycle of a group of people in the area, it is possible to translate the tricot in oneself, to be composed of 6 elements: turning on the kuti near the point and symmetry, three straight lines.

The stench of the images in fig. 1 with red lines. The elements of the self-alignment group of the correct tricycle can be set and given. Let's explain, numbered the vertices of the correct tricycle with the numbers 1, 2, 3. Be it self-alignment of the tricycle, translate points 1, 2, 3 into the same points, or taken in the same order, so that it can be cleverly written in the view of one of these :

etc.

The numbers 1, 2, 3 denote the numbers of the vertices, in which the vertices 1, 2, 3 go over in the result of the opened ruch.

Projective space and models.

Understanding the design space and model of the design space. Basic facts of projective geometry. The link straight with the center at point O is a model of the projective area. Projective points. Expanded area - model of projective area. Expanding the trivial Afinny or Euclidean space is a model of the projected space. Image of flat and spacious figures with parallel design.

Understanding the design space and model of the design space:

The design space over the field is a space that can be laid out in straight lines (of the same size) of some linear space over the given field. Straight to the open space dots projected space. The price of the appointment is to be visited on a large number of occasions

As there is a small dimension, then the dimension of the projective space is called a number, and the projective space itself is known and called an associative z (well, it’s accepted, it’s meaningful).

Moving from the vector space to the general projective space to be called project_visation spaciousness.

Points can be described beyond one-sided coordinates.

Basic facts of projective geometry: Projective geometry is a division of geometry, which provides space for design areas. The main specialty projective geometry of the polarity in the principle of subordination, which gives vitality to the symmetry in a rich design. Projective geometry can be viewed both from a purely geometric point of view, also from an analytical point (beyond the one-sided coordinates) and salgebraic, showing the projective area and structure over the field. Often, and historically, a speech projective area is viewed as the Euclidean area with the addition of "straight in the absence".

Todi yak power figures, with yakim I can help Euclid's geometriya, є metric(The specific values of the kutiv, vidrizkiv, area), and the equivalence of figures is equal congruence(That is, if the figuri can be transferred alone to the one for the help from the protection of the metric authorities) geometric figures, They are saved when re-making more zagalny type, Ніж Рх. Projective geometriya is engaged in the development of the authorities of the figur, in the class project revisions, And also the tsikh perversion themselves.

Geometry is projective in addition to Euclidean, it is a beautiful and simple solution for building buildings, accelerating the presence of parallel straight lines. Particularly simple and vitonized is the projective theory of final retinues.

Є three main approaches to projective geometry: independent axiomatization, additional Euclidean geometry, and structure over the field.

axiomatization

The design space is possible due to the additional set of axioms.

Coxeter nadaє nasty:

1.Isnuє straight and not speckled.

2. On the skin line є take three points.

3. Through two points it is possible to draw exactly one straight line.

4. Yaksho A, B, C, і D- різні points і ABі CD change, then ACі BD overhaul.

5. Yaksho ABC- area, then there is one point not in the area ABC.

6. Two different areas are overturned at least in two points.

7. Three diagonal points of the main chotirikutnik are NOT collinear.

8. There are three points on a straight line X X

The projective area (without the third vimir) is indicated by cheap axioms:

1. Through two points you can draw exactly one straight line.

2. Be straightforward.

3. There are many points, there are not three collinear points.

4. Three diagonal points of the upper chotiricutniks are NOT collinear.

5. There are three points on a straight line X invariant according to the relation to the projectivity φ, then all points on X Price invariant up to φ.

6. Desargues's theorem: If two tricycles are promising to the point, then the stench is promising to the straight.

If the third way is evident, Desargues' theorem can be brought without introducing ideal points and straight lines.

Expanded area - model of projective area: in the affine spaciousness A3, the link is straight S (O) centered at point O and the area Π, which does not pass through the center of the link: O 6∈ Π. The link is straight into the affine space є the model of the projective area. It is set to represent the powerless points of the area Π on the soundless straight calls S (Fuck, praying for food, forgive me)

Expanding trivimirne Afinny or Euclidean space - a model of the projected space:

In order to make the image surreal, we repeat the process of formal extension of the affine area Π to the projective area, Π, additionally the area Π without any non-strong points (M∞) such that: ((M∞)) = P0 (O). Oscillations in the preimage of the skin area of the link of the areas S (O) є are straight on the area d, then it is obvious that all non-strong points of the extended area are without any problem: Π = Π ∩ (M∞), (M∞), ∞ extended area, which is the preimage of the special area Π0: (d∞) = P0 (O) (= Π0). (I.23) Being at home, but not being equal P0 (O) = Π0 here and above we will be sensible in the sense of the equalness of many points, albeit more overwhelming with a different structure. Having supplemented the affine area with unchanged straight lines, we desired that the image (I.21) became bijective at all points of the expanded area:

Image of flat and spacious figures with parallel design:

The stereometry vivchayutsya spacious figurines, the protest on the chair stench appears in the view of flat figurines. What is the rank of the image of the vast figure on the square? Call in geometriya for tsyo vikorystvua parallel to the project. Come on p - deyaka area, l- overflowing straight (Fig. 1). Through a certain point A Don't be straight l, Conducted straight, parallel to straight l... The point overflowing the center straight from the area p is called the parallel projection of the point A square p in a straight line l... meaningfully її A". Yaksho point A lay straight l, Then a parallel projection A on the area p, the point of overturning straight l with area p.

In such a rank, skin point A open space for a projection A"On the area p. The number of projects is called parallel projects on the area p in a straight line l.

A group of design revisions. Dodatok to the solution of tasks.

Understanding the design re-development of the area. Put on the design reworking of the area. The power of the design revisions. Homology, power of homology. A group of design revisions.

Understanding the design re-development of the area: Understanding of the project re-implementation of the public understanding of the central projection. Also, the viconati is located from the central projection of the area α to the deyak area α 1, instead of the projection α 1 to α 2, α 2 to α 3, ... n I know for α 1, then the composition of all the projections and the projective transformation of the area α; in such a lance you can include parallel projections.

Attach the design re-creation of the area: The projective re-incarnations of the new area are called self-imaging in a mutually unambiguous way, when the collinearity of the points is taken care of, or, in other words, be a straight line. Be-like a projective re-enactment є composition of a lance of central and parallel projections. Afinny re-creation is a concept of the project, with which it is not endlessly straight forward to pass in itself.

The power of the project revisions:

When the design is redesigned, three specks do not lie in a straight line; do not lie in a straight line into three specks.

When the project is re-created, the benchmark go to the benchmark.

With the design re-implementation, the straight line goes into the straight line, the beam goes into the beam.

Homology, power of homology:

Projectively, the transformation of the area, as there is a straight line of invariant points, and hence, a bunch of invariable straight lines is called homology.

1. Straight, so pass through nezbіzhnі vіdpovіdnі points of homology, і invarіant straight lines;

2. Straight, passing through non-negligible homology points, lying on one bundle, the center of which is an invariable point.

3. Point, її image and center of homology lie on one straight line.

A group of project revisions: The projective image of the projective area P 2 on itself is visible projectively, so that the projective reconstruction of the whole area (P 2 '= P 2) is not visible.

Yak and earlier compositions f design rework f 1 і f 2 projective area P 2 is called the result of the last revision f 1 і f 2: f = f 2 ° f 1.

Theorem 1: without all the design revisions of the projective area P 2 є by a group of composites of design revisions.